Complete
The Square
(x+4)(x-3)=0
FOIL
X2 – 5x +4
CRASH COURSE IN
QUADRATICS
In preparation for the Algebra CST
Multiplying Polynomials
Area Model of Multiplication
To multiply 68 x 34:
•Write the two numbers in expanded notation
and multiply one box at a time.
•After you have multiplied the numbers, add
all of the products together.
60
30
+
4
+
8
(30)(60)
1800
(30)(8)
240
(4)(60)
240
(4)(8)
32
1800+240+240+32=2312
Now you try one… 48 x 53
Multiplying Polynomials
Area Model of Multiplication
To multiply (x+2)(x+3):
•Write the two numbers in expanded notation
and multiply one box at a time.
•After you have multiplied the numbers, add
all of the products together.
x
x
+
3
+
2
(x)(x)
x2
(x)(2)
2x
(3)(x)
3x
(3)(2)
6
X2 + 5x + 6
Now you try one…
(x+5)(x+1)
Multiplying Polynomials
FOIL
( x + 2 ) ( x + 3)
First
(x)(x) = x2
Outer
(x)(3) = 3x
Inner
(2)(x) = 2x
Last
(2)(3) = 6
Combine like terms…
= x2 + 5x + 6
Multiplying Polynomials
x2 + 5x + 6
a=1
b=5
c=6
ax2 + bx + c
Factoring Polynomials
10
12
3
4
2
5
7
7
3
6
5
6
6
14
1
7
2
7
9
18
21
9
10
4
Ask yourself… “What two numbers
multiplied together give you the top
digit and added together give you
the bottom?”
Factoring Polynomials
X2
+ 7x + 12
X2 + 13x + 36
X2 - 6x - 40
12
7
36
13
-40
-6
(x +
)(x+
)
(x +
)(x+
)
(x +
)(x+
)
Perfect Square Trinomial
X2 + 12 + 36
X*X
(x + 6)2
6 * 6
X2 -- 14 + 49
X*X
(x + 6)(x + 6)
7 * 7
(x - 7)(x - 7)
(x - 7)2
Solving Quadratic Equations
•Graphing
•Factoring
•Using Square Roots
•Completing the Square
•Quadratic Formula
Graphing Quadratic Equations
x2 – 4x = 0
x
y=x2 - 4x
y
x, y
0
02 – 4(0)
0
0, 0
2
22 - 4(2)
-4
2, -4
4
42 – 4(4)
0
4, 0
The Solution is the
________________
Find the solution for each graph:
Factoring Quadratic Equations
Using the Zero Product Property
(x-3)(x+7)=0
(x-3)=0
x=3
(x+7)=0
x = -7
Factoring Quadratic Equations
Solve using the Zero Product Property
(x-3)(x+4)=0
Can you solve in your head?
(x-2)(x+1)=0
(x+3)(2x-8)=0
(3x-1)(4x+1)=0
(3x+1)(8x-2)=0
x2
+ 12x + 36
x2 - 21x = 72
-72
-21
x=
x=
If x2 is added to x, the sum is 42. What
are the values of x?
Using Square Roots
Square-Root Property
x2 = 16
4x2 – 25 = 0
+25 +25
√x2 = √16
√4x2 = √25
x = +4
2x = 5
2
2
x2 = 16
(4)2= 16
(-4)2 = 16
x = + 2.5
4x2 = 25
Completing the Square
Using Algebra Tiles
x2 + 6x
a= 1 b=6
b
2
( )
x2 + 6x = 0
+9 +9
2
b=6
c=0
x2 + 6x + 9 = 9
(x+3)(x+3)=9
(x+3)2 = 9
√(x+3)2 = √ 9
x+3 = + 3
2
( )
( 62 )
6
2
x+3= 3
x=0
x+3= -3
x = -6
2
=9
Completing the Square
Add to both
sides of the
equation
x2
Factor
the
Perfect
Square
+ 14x = 15
+ 49 + 49
2
(142)
b = 14
= 72 =49
x2 + 14x + 49 = 64
(x+7)(x+7)=64
(x+7)2 = 64
√(x+7)2 = √ 64
x+7 = + 8
x+7= 8
x=1
x+7= -8
x = -15
Completing the Square
3x2 – 10x = -3
3 3
3
Reduce
x2 - 10x = -1
3
Factor
the
Perfect
Square
x2 - 10x = -1
3
+25 +25
9
9
x2 - 10x + 25 = 16
3 2 9
9
x–5
16
=
3
9
(
b = 10
3
√(
x–5
3
) =√169
3
2
2
= 100
36
25
9
Add to both
sides of the
equation
-9 + 25 = 16
9
9
9
)
2
(-10 * 1)
(x – 53 )=+ 43
x–5=4
3 3
x=9
3
x – 5 = -4
3 3
x=1
3
Completing the Square
x 2 - 8x = 12
x 2 - 8x = 5
2
x + 4x = 6
2
x - 4x = 8
2
ax – bx = c
What should be added to both sides of this equation?
The Quadratic Formula
x2 + 5x + 6
2x2 + 3x – 5 = 0
ax2 + bx + c
x=
-b + √ b2 – 4ac
2a
x=
-b + √ b2 – 4ac
2a
a=1
b=5
c=6
ax2 + bx + c
a = 2 b = 3 c = -5
x=
-3 + √ 32 – 4(2)(-5)
2(2)
x=
-3 + √ 9 – (-40)
x=
-3 + √ 49
4
4
x=
-3 + 7
4
x=
-3 + 7
4
x=4
x=
-3 - 7
4
x = - 2.5
The Quadratic Formula
2x = x2 - 3
ax2 + bx + c
0 = x2 – 2x - 3
x=
2x = x2 - 3
-2x
-2x
a = 1 b = -2
ax2 + bx + c
-b + √ b2 – 4ac
x=
2a
x=
-(-2) + √ (-2)2 – 4(1)(-3)
x=
x=
2
2+4
2
x=
2+4
2
x=3
0 = x2 – 2x - 3
c = -3
-(-2) + √ (-2)2 – 4(1)(-3)
2 + √ 4 +12
2
2 + √ 16
2a
2(1)
2(1)
x=
x=
-b + √ b2 – 4ac
x=
2-4
2
x = -1
x=
2 + √ 16
2
Solving Quadratic Equations
•Graphing
•Factoring
•Using Square Roots
•Completing the Square
•Quadratic Formula
Solving Quadratic Equations
2
x + 4x - 2 = 0
2
x - 5x + 4 = 0